An antenna is generally a transducer device that receives or transmits electromagnetic radiation. The frequency of radiation that may be received or transmitted by an antenna is dependent on the size of the antenna, the speed of light and the distance that electrons can travel (electron mobility) in the material medium of the antenna. Because electromagnetic waves propagate more slowly in a medium than in free space, the same number of waves will span a greater distance in free space than in the transmission medium, hence the transmission medium is said to have an electrical length that is greater than its physical length. Typically, the electrical length of an antenna is expressed in units of the wavelength (in the antenna medium) corresponding to the resonant frequency of the antenna.
Antennas are typically associated with signals having a frequency of about 30 kHz to about 30 GHz and may be associated with, for example, longwave AM radio broadcasting, RFID tags, wireless LAN, radars, and satellite television broadcasting. In general, the electrical length of an antenna is on the order of the free-space wavelength of the radiation at which the antenna is resonant. For example, a dipole antenna is typically about ¼th the free-space wavelength. Similarly, the physical length of an antenna is on the order of the wavelength in the antenna medium of the radiation at which the antenna is resonant. Given that the wavelength of electromagnetic radiation is shorter in a medium than in free space, the physical length of an antenna is typically shorter than its electrical length.
Every antenna has a characteristic impedance, which is the ratio of voltage to current at any given point in the antenna. In general, the impedance of an antenna is a complex number dependent on frequency of the voltage (or current). The real part of the complex impedance is pure resistance and is frequency independent. The complex part (also called reactance) is the frequency dependent part of the impedance and may be either directly proportional to the frequency (inductive reactance) or inversely proportional to the frequency (capacitive reactance). The resonant frequency of an antenna is defined as the frequency at which the capacitive impedance and the inductive impedance of the antenna are equal and opposite to each other, thereby cancelling each other and making the impedance at that frequency purely resistive. The voltage and current at this frequency are in phase with each other.
The complex impedance Za, of an antenna may be determined by the following formula:Za=Ra+iXa  (1)where Ra is the resistance, and Xa is the reactance of the antenna, having a capacitive component and an inductive component according to:Xa=XC+XL=(−1/ωC)+ωL  (2)where ω=2πf is the angular frequency, and f is the frequency. It is evident that by changing one or both of the inductive impedance XL and the capacitive impedance XC that the resonant frequency of an antenna can be changed.
For optical frequencies, ranging from terahertz to petahertz, metals are not perfect conductors but may be described as free-electron gases. Incident radiation at these frequencies is not perfectly reflected, but rather penetrates the metal surface and produces oscillations in the free-electron gas. Quantum effects apply at such frequencies, and surface plasmon resonances cause deviations in material properties. Classical antenna theory, therefore, needs to be modified by replacing classical impedance with local density of electromagnetic states (LDOS). The LDOS can be expressed in terms of Green's function tensor . For a quantum dipole {right arrow over (p)} located at {right arrow over (r)}0 the partial LDOS is expressed as:
                                                        ρ              p                        ⁡                          (                                                                    r                    0                                    →                                ,                ω                            )                                =                                                    6                ⁢                ω                                            π                ⁢                                                                  ⁢                                  c                  2                                                      ⁡                          [                                                                                          n                      p                                        →                                    ·                  Im                                ⁢                                                      {                                                                  G                        ↔                                            ⁡                                              (                                                                                                            r                              0                                                        →                                                    ,                                                                                    r                              0                                                        →                                                    ,                          ω                                                )                                                              }                                    ·                                                            n                      p                                        →                                                              ]                                      ,                            Eq        .                                  ⁢                  (          1          )                    where {right arrow over (np)} is the unit vector in the direction of the dipole {right arrow over (p)}, and ω is the angular frequency. The full LDOS can be obtained by averaging the partial LDOS of Eq. (1), and is expressed as:
                                                                        ρ                ⁡                                  (                                                                                    r                        0                                            →                                        ,                    ω                                    )                                            =                            ⁢                              〈                                                      ρ                    p                                    ⁡                                      (                                                                                            r                          0                                                →                                            ,                      ω                                        )                                                  〉                                                                                                        =                                ⁢                                                                            2                      ⁢                      ω                                                              π                      ⁢                                                                                          ⁢                                              c                        2                                                                              ⁢                                      Im                    ⁡                                          [                                                                        T                          r                                                ⁡                                                  (                                                                                    G                              ↔                                                        ⁡                                                          (                                                                                                                                    r                                    0                                                                    →                                                                ,                                                                                                      r                                    0                                                                    →                                                                ,                                ω                                                            )                                                                                )                                                                    ]                                                                                  ,                                                          Eq        .                                  ⁢                  (          2          )                    where Tr denotes the trace.
By representing the quantum emitter as a classical dipole {right arrow over (p)}, located at {right arrow over (r)}0, the power dissipated by the emitter at angular frequency ω is expressed as:
                                                        P              =                            ⁢                                                1                  2                                ⁢                                                      ∫                    V                                                                                                  ⁢                                      Re                    ⁢                                          {                                                                        j                          →                                                ·                                                  E                          →                                                                    }                                        ⁢                                                                                  ⁢                                          ⅆ                      V                                                                                                                                              =                            ⁢                                                ω                  2                                ⁢                Im                ⁢                                  {                                                                                    p                        →                                            *                                        ·                                                                  E                        →                                            ⁡                                              (                                                                              r                            0                                                    →                                                )                                                                              }                                                                                                        =                            ⁢                                                                    πω                    2                                                        12                    ⁢                                          ɛ                      0                                                                      ⁢                                                                                                p                      →                                                                            2                                ⁢                                                      ρ                    p                                    ⁡                                      (                                                                                            r                          0                                                →                                            ,                      ω                                        )                                                                                                          Eq        .                                  ⁢                  (          3          )                    where, V is the source volume, {right arrow over (j)} is the current density, and {right arrow over (E)} is the electric field.
Using the expression for dipole radiation in free space: P0=|{right arrow over (p)}|2ω4/(12π∈0c3), LDOS in terms of normalized radiation can be expressed as:
                                          ρ            p                    ⁡                      (                                                            r                  0                                →                            ,              ω                        )                          =                                            ω              2                                                      π                2                            ⁢                              c                3                                              ⁢                                    P                              P                0                                      .                                              Eq        .                                  ⁢                  (          4          )                    
The antenna resistance can then be calculated as:
                    R        =                              π                          12              ⁢                              ɛ                0                                              ⁢                                                    ρ                p                            ⁡                              (                                                                            r                      0                                        →                                    ,                  ω                                )                                      .                                              Eq        .                                  ⁢                  (          5          )                    